| L(s) = 1 | − 0.792·2-s + 3-s − 1.37·4-s − 0.792·6-s − 2.52·7-s + 2.67·8-s + 9-s − 1.37·12-s − 6.78·13-s + 2·14-s + 0.627·16-s − 6.63·17-s − 0.792·18-s − 7.72·19-s − 2.52·21-s − 8·23-s + 2.67·24-s + 5.37·26-s + 27-s + 3.46·28-s − 3.16·29-s − 3.37·31-s − 5.84·32-s + 5.25·34-s − 1.37·36-s − 5·37-s + 6.11·38-s + ⋯ |
| L(s) = 1 | − 0.560·2-s + 0.577·3-s − 0.686·4-s − 0.323·6-s − 0.954·7-s + 0.944·8-s + 0.333·9-s − 0.396·12-s − 1.88·13-s + 0.534·14-s + 0.156·16-s − 1.60·17-s − 0.186·18-s − 1.77·19-s − 0.550·21-s − 1.66·23-s + 0.545·24-s + 1.05·26-s + 0.192·27-s + 0.654·28-s − 0.588·29-s − 0.605·31-s − 1.03·32-s + 0.901·34-s − 0.228·36-s − 0.821·37-s + 0.992·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.04357641748\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04357641748\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.792T + 2T^{2} \) |
| 7 | \( 1 + 2.52T + 7T^{2} \) |
| 13 | \( 1 + 6.78T + 13T^{2} \) |
| 17 | \( 1 + 6.63T + 17T^{2} \) |
| 19 | \( 1 + 7.72T + 19T^{2} \) |
| 23 | \( 1 + 8T + 23T^{2} \) |
| 29 | \( 1 + 3.16T + 29T^{2} \) |
| 31 | \( 1 + 3.37T + 31T^{2} \) |
| 37 | \( 1 + 5T + 37T^{2} \) |
| 41 | \( 1 - 6.92T + 41T^{2} \) |
| 43 | \( 1 + 9.94T + 43T^{2} \) |
| 47 | \( 1 + 2.74T + 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 2.67T + 61T^{2} \) |
| 67 | \( 1 + 6.11T + 67T^{2} \) |
| 71 | \( 1 + 0.744T + 71T^{2} \) |
| 73 | \( 1 + 5.19T + 73T^{2} \) |
| 79 | \( 1 + 0.147T + 79T^{2} \) |
| 83 | \( 1 - 1.87T + 83T^{2} \) |
| 89 | \( 1 + 17.4T + 89T^{2} \) |
| 97 | \( 1 - 3.62T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.80249653896162231380920184441, −7.16094849275382881948016584935, −6.60016766105726005045671360950, −5.71428375770381688300030498108, −4.70398829981743789645496452258, −4.25879365006697737128370813263, −3.53465375237912974058674869952, −2.32913189735522683948155299874, −1.98602178282388334004510061181, −0.098829654736545109562797906734,
0.098829654736545109562797906734, 1.98602178282388334004510061181, 2.32913189735522683948155299874, 3.53465375237912974058674869952, 4.25879365006697737128370813263, 4.70398829981743789645496452258, 5.71428375770381688300030498108, 6.60016766105726005045671360950, 7.16094849275382881948016584935, 7.80249653896162231380920184441
